"Timothy Y. Chow" <tchow at alum.mit.edu> wrote:
>I'm not sure exactly what kind of narrowing you are
>asking for. My
>specific suggestion is that the fundamental
>difference between
>justification in mathematics and justification in
>other areas of knowledge
>is that a unit of mathematical knowledge (i.e., a
>theorem) can be
>*definitively* justified by a *bounded* amount of
>evidence (i.e., a
>proof), provided we assume that the evidence is
>error-free. In contrast,
>no bounded amount of evidence can ever definitively
>justify, say, a law of
>physics, even if we grant by fiat that the evidence
>Is my assertion not sufficiently narrow? Why not?
I do not think the nature of justification in other
fields of science is in the unbounded nature of
possible proofs (or warrant mechanisms)... if you want
to speak of unbounded number of possible proofs, then
also in Mathematics too the possibility is not
eliminated. Other sciences too use mechanisms of
generalization. It is possible to write nice bounded
proofs for bodies of vague and imprecise knowledge and
those become part of the branch of science in
question. This trend is bound to become more intense
with time. So much so that we may as well be able to
speak Mathematics in say some part of the cognitive
sciences.
Apparently you mean "potentially bounded", rather than
"bounded"... then also "so many Mathematicians have
been wasting their time in doing non-Mathematics like
trying to solve apparently unsolvable problems".
But however if we look closely at the different
universes of mathematical discourse, then we can
abstract features that will provide at least some
levels of the desired contrast. In particular the
so-called Platonic universe has many of these
features. Since you want to approach the issue by way
of proof or rather justification (the two are
different), it will help if the possible features are
captured into it by way of intentionality or possibly
by restriction to particular isms.
As you have put it ... it looks too open.
Is the intended concept of proof a 'formal one within
a language'?
Is your Mathematics, the Mathematics of which Platonic
universe ?
For which isms in Mathematics is your proposal
intended for?
Is your proposal totally against 'applied mathematics'
being part of Mathematics ?
Computer scientists write bounded proofs?
Best
A. Mani
--
A. Mani
Member, Cal. Math. Soc
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